$...$ correspondences of an equilateral triangle with itself are similarity.

In $\Delta ABC$,$m \angle A + m \angle C = m \angle B$ and $AC : AB = 17 : 15$. If $BC = 8$,then the area of $\Delta ABC$ is.............

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QS}$ is an altitude to the hypotenuse $PR$. If $PQ = 6$ and $PS = 4$,find $QS$,$QR$,and $RS$.

The areas of two similar triangles are $200$ and $128$. Then,the ratio of their corresponding altitudes is.....

The length of a diagonal of a square is $8 \sqrt{2}$. Then,the perimeter of the square is..............

$A$ person begins his journey from point $A$. From $A$,he moves $800 \, m$ towards north to reach point $B$. From $B$,he moves $500 \, m$ towards east to reach point $C$. From $C$,he moves $400 \, m$ towards north to reach point $D$. Find the direct distance in $m$ from $A$ to $D$.